So how do I play?
Playing is as easy as creating nuclear fusion using only the power of your mind, much like any roguelike game, so I'll offer a brief overview of your beginning game so you have some idea how to play (although there is also a reasonably good documentation in-game.)
First off, the controls. Most of everything in the game is controlled by the +- keys, the enter key, the space keys, and the arrow keys. Usually, the +- keys will move around in the command menu, the enter key will select something, the space key will back out of whatever menu you are in, and the arrow keys will move you around in the real map; but it's not 100% consistent. Be prepared to experiment.
You start off creating a world. Since every world's features, civilizations, legends and so on are randomly generated, no one is playing the same world. On the other hand, world generation can take a long time to ensure that you have a world capable of including everything; I've seen results anywhere between 75 and 175 rejections before it generated something. It will also take some time to generate the actual world; the landscape is fast, but civilization building and unit building is fairly slow. Thankfully, you will not have to do this often; each world contains 50 spots for Fortresses, and a large, large number of things to kill in Adventurer Mode.
Once you've done this, you can start a new game. There are three modes: Dwarf Fortress is the meat of the game, the city-building sim where you command dwarves. Adventurer Mode is the token roguelike, which currently consists of wandering through the world with whatever weapons you have until you get killed by monsters, or talking to people to find out the legends of the world, which then show up in Legends mode, which is basically an overview of what all has happened in your world (including a lot of stuff that only happened in the world gen portion); however, it's all secret at the start, so you have to find out about it in Adventurer Mode for it to show up.
Dwarf Fortress mode starts you out in the preparation screen. You can either select to Play Now!, which gives you a reasonable set of skills and equipment at a random starting spot, or prepare for the journey, which allows you to pick out all of this yourself with a pool of 200 points.
First you work out your dwarves. You won't need every skill at first, but you want people who can mine, chop wood and make wooden things, cut stone to make stone things, farm fields, fish, possibly hunt (but if you do take someone who can hunt, make sure to give him a weapon skill, ideally Marksdwarf.) However, any dwarf can learn anything if he's assigned the job, and their skills will increase as they use them, so don't feel the need to buy every skill in the list.
Next you'll need supplies. I myself have never been quite clear on what kinds of supplies are best; I usually increase my food supply by a bit and add in the other two types of seeds to ensure I can plant anything I might want later on, rather than having to rely on finding the seeds. You may also want to add in another pick, axe; or possibly weaponry, depending on how dangerous your starting area is. (Picks are the tool for mining, axes are the tool for wood chopping.)
Finally, you select your area. This costs no points, so pick a reasonably temperate area with good forestation, vegetation, and not too many things trying to kill you. You can pick a more dangerous area for greater challenge, but you run the risk of getting killed or not having enough resources, or countless other problems.
So, assuming you've done all this, all that remains is to hit e to embark and send your dwarves on their merry little way.
Not Dying in a variety of colorful ways
So, you're in front of a cliff face with a bunch of faces milling around a nine square of brown blocks (your wagon.) Time to start! First hit space to pause, so nothing stupid happens while you're telling your dwarves what to do. Then, hit TAB a few times to make the interface less annoying; you'll probably want to keep the command list up for now, but the world map is unimportant.
Next hit d. This will bring up your Designations list, for things like telling your dwarves what parts of the mountain to hollow out. Your cursor should be a yellow block; move it over to the mountain and pick a spot connecting with the outside, and designate an entrance to your soon-to-be fortress of evil. While you're at it, designate some trees for your carpenter to chop down in the same way, but don't designate too many; you want him to also have some free time to make stuff from those trees.
Now, exit out of that. Now hit p. This will bring up your stockpile designation list, which tells your dwarves where to put all the various trash that they find. Since it's only a designation, no one has to build these; they just pop into existence. Put down a Food, Wood, Stone, and Refuse designation at the bare minimum. Your stone designation should be a good size, as well, since you'll be pulling out a LOT of rock from your mining efforts.
Now, exit out of that and unpause. Your dwarves should get to work, moving stuff around, digging, and chopping trees. Watch them work uncomplainingly at back-breaking labor for no pay. Possibly giggle evilly. Once you've got some rocks, though, pause again and hit b. This will be one of your most visited windows; this is where you place things.
For now, you don't have most of the things to place, so find Workshops (or hit w from within this menu). Select a Mason's Workshop; it should go to a list of rock types you have. Probably short right now, but later this will be massive. Hit enter to pick any random type (use rocks, and not wood; wood is harder to come by, and you want to save it for other projects.)
Now you should have a 3x3 square of green Xs. The light green Xs indicate parts of the workshop that will be passable once it's complete, while the dark green Xs indicate parts that will be solid. For now, place it outside somewhere. Your mason dwarf should pick up a rock and lug it over to the workshop, and it should get finished.
Repeat the above process with a Carpenter's workshop now. You should note however that this will probably not finish anytime soon, since your carpenter has to build it, and he's probably busy cutting his wrist down trees. The various types of buildings take various skills to make; usually, whatever profession will be using the workshop is needed to build it.
Let's take a quick look at how you give dwarves new jobs. Go back to the main menu and hit v. This should give you a cursor, and your window should become an examination of whatever dwarf is closest to that cursor. You can hit various buttons within this, such as g to see general stats, i to see inventory, p to see his settings, and w to see his condition. Go to his settings for now.
There, you should be able to hit l for Labor, and a list of job types will pop up. You can scroll through it with + and -, and hit enter to select/deselect any jobs you want; he will then do or not do those jobs. These jobs include just about every task a dwarf could need to do, from refuse hauling to metalworking.
Now, you should have at least a Mason's Workshop done, so it's time to do something. Go back to the overview and hit q, which should bring up a cursor and change the window to the build orders for whatever building is closest. Find your Mason's Workshop, and hit a. This will bring up a list of things he can build. Right now, you want to add Stone Blocks first. You'll need other things, too; right now, you need tables and thrones (which are basically rock chairs.)
Build one throne per dwarf you have, and a table for each one (each table can have up to four thrones at it, but if you push them next to each other to make one long table then that'll go down.)
Next, dig out another room, at least 3x5. You'll put the finished tables and thrones in here, and it will become your dining room. Dwarves dislike eating without sitting down, and that will become an unhappy thought, which in turn pushes them closer to being unhappy, which is bad on many levels.
Now wait. Your dwarves should cheerfully go about their work. Once your Carpenter's Workshop is done, queue up a bucket and one bed per dwarf you have, then dig out another room at least 3x4 to put the beds in. Later on you'll need to build a room for each dwarf, but early on they're willing to sleep barracks style.
Once your beds, bucket and blocks, or tables/thrones are done, hit b again. You can place everything, not just buildings, from this menu. If your tables are done, plop them down in whatever room you made your dining room, in a pattern such that the thrones are next to the tables. If your beds are done, plop them down in your barracks. Finally, if your blocks/bucket are done, pick out Well from the building menu and put it in an open room with lots of access, since your dwarves will come here regularly.
Once all of these are done, you should have a basic fortress which won't die within five minutes from berserk dwarves, and where you go from there is up to you. I'll give you two more important things to do, however.
The first is a Craftsdwarf's Workshop. Here is where you will build all kinds of useful and neat things, primarily trade goods. These will go in a Finished Goods pile. They're important because anything you didn't start with the ability to produce, and a lot of things you did, you'll want to get from the rest of the world (for which you'll need a trade depot, and also a road for trading with non-dwarven traders.) In fact, if you're clever and have been reading the text in the game, you'll have noted that a supply caravan is supposed to arrive in the fall; but their goods aren't free, so you'd better have some stuff to trade back to them if you want their supplies.
The second part is farming. This is complex and confusing and important, so it gets special mention. You see, you farm in-cave; apparently you farm mushrooms or some other plant that doesn't require sunlight. To do this, however, you'll need to find your first in-cave river. Make a long, 1xlots dig designation into the depths of the mountain. You should find one after a few screens; in fact, your dwarf will probably have to run for his life from the flooding after you hit the river. Once the flooding recedes, however, you can open up holes to the river more or less safely.
This is important, since your dwarves farm with river mud. Dig open a spot next to the river; keep it separate from the rest of the cave, accessible only by a one square entry, where you'll put a door (build it from the Mason's Workshop.) Now, dig open one or two squares to the river. Nothing should flood at this point.
Now, you'll need to dig out a small adjunct room next to the door to the river-access room, which we'll call the control room. Once the door is between the access room and the control room, hit q and you should be able to set up the door. Keep it closed, but not locked.
Now, for the next trick, you will need to build a floodgate from your mason's workshop, and a mechanics workshop, from which you'll need three mechanisms. Make one of your dwarves work at Mechanics if you don't already have one.
Put your floodgate at the spot in the access room where the room meets the river. Next, build a Lever from b; this will take one mechanism and will be under Traps/Levers. Once the lever is built, hit q again and you should be able to set it up. Add the task "Hook up to a floodgate; it should give a list of floodgates, and there's probably only the one, so select it. Then it'll need two mechanisms; one for the gate, and another one for the lever.
EDIT: There've been some problems with water pathing recently, warranting an edit here: It may be necessary to install a second floodgate in front of your first one to convince the water to enter your access room, and some have suggested using a door and a second lever at the junction between access and control rooms, for additional safety. Note that every floodgate you hook up will require two mechanisms (one at the gate, one at the lever), and levers require one mechanism to construct.
Once all of this is set up, go back to the lever's menu and add the task "Pull the lever." One of your dwarves should come up to the lever soon and pull it, at which point the access room will be flooded (and the control room as well if you didn't set the door to be closed at all times.) Pull the lever again, and the water will recede.
But wait... the rocks in the access room are now brown with river mud! You can now build Farm Plots on it from the building menu. Once the plot is built, you can access it's menu the same way as with any building, and select what you want planted there; plump helmets are a good start, since they're a normal food item.
Every season you'll need to re-indicate what you want planted there, and the mud will harden in the winter, so you can't plant then; you'll need to re-flood the room in the spring, as well.
However, this is your primary food source by far; hunting is slow and unreliable, especially since the hunt target code is RETARDED (Well there's a deer right in front of the fortress, but let's go to THE END OF THE MAP and kill there instead!), and fishing only provides one fish/use and takes time; it also demands the full attention of the fisher, whereas one farmer can fill up a small plot on his own and get 3 plump helmets per spot.
So what else is there?
There's a lot more to the game as well, including a lot I haven't seen myself; it's only been out since Tuesday, and is only an alpha right now (although a lot of the core functionality is there; there's just a lot of missing small things.)
Increasing Social Dynamics!
Your dwarves live in an egalitarian, communist society at first; everyone has a few personal possessions, but everything is shared between people and whatever is needed is a tool "for the people." However, as your fortress expands, nobles will move in; they demand a lot, but in return, they expand your prestige and social dynamic. For example, the Manager allows you to dictate orders of things you need built, rather than queuing things at each workshop; with him comes the Sheriff, and criminal justice. Supposedly, later on once you start making money, you can even have an internal economy.
Wars and Monsters!
Those military skills aren't just for deer. Monsters will sometimes show up in your fortress, both from outside and from inside. You'll need military dwarves to defend you from frogmen using your dwarves as farm creatures and raiding you from your cave river, and also to keep jealous goblin, human, or elven kingdoms from outside from entering your base and killing your mans. This is especially true if you live near Evil regions, special parts of the map which are corrupted by evil (think the area around Mt. Doom from LOTR. Nothing good or friendly lives there.)